Generalized Differentiation in Wasserstein Space and Application to Multiagent Control Problem

Multi-agent systems consist of numerous interacting agents. The sheer number of agents is so vast that it becomes impractical to describe the dynamics of each one individually. Consequently, to characterize the evolution of such systems, problems are typically addressed within the Wasserstein space of probability measures: indeed, the state is represented by a probability measure expressing the distribution of the agents. Several concepts of generalized differentiation in the Wasserstein space of probability measures have been proposed in the literature in order to deal with the intrinsic nonsmoothness arising in the context of optimization problems in such spaces, mainly in the framework of mean field games, mean field control and population dynamics. In this paper we introduce a concept of admissible variation encompassing some of the most popular definitions as special cases, proving that under suitable commonly used assumptions they are all equivalent, and using it to derive a comparison principle for viscosity solutions of an Hamilton-Jacobi-Bellman equation arising in an optimal control of a multiagent systems of leader-follower type.